Approximate smooth solutions of a mathematical model for the activation and clonal expansion of T cells

D. Criaco; M. Dolfin; L. Restuccia

doi:10.3934/mbe.2013.10.59

Article Contents

2013, Volume 10, Issue 1: 59-73. Doi: 10.3934/mbe.2013.10.59

This issue Previous Article Distributed delays in a hybrid model of tumor-Immune system interplay Next Article An agent-based model for elasto-plastic mechanical interactions between cells, basement membrane and extracellular matrix

Approximate smooth solutions of a mathematical model for the activation and clonal expansion of T cells

1.
Department of Mathematics and Informatics, University of Messina, Viale F. Stagno d'Alcontres n.31, 98166 Messina
2.
Department I.C.I.E.A.M.A., University of Messina, Contrada Di Dio (S.Agata), 98166 Messina

Received: April 2012

Revised: September 2012

Published: December 2012

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Abstract

In a previous paper a mathematical model was developed for the dynamics of activation and clonal expansion of T cells during the immune response to a single type of antigen challenge, constructed phenomenologically in the macroscopic framework of a thermodynamic theory of continuum mechanics for reacting and proliferating fluid mixtures. The present contribution deals with approximate smooth solutions, called asymptotic waves, of the system of PDEs describing the introduced model, obtained using a suitable perturbative method. In particular, in the one-dimensional case, after deriving the expression of the velocity along the characteristic rays and the equation of the wave front, the transport equation for the first perturbation term of the asymptotic solution is obtained. Finally, it is shown that this transport equation can be reduced to an equation similar to Burgers equation.

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Mathematics Subject Classification: Primary: 92B05; Secondary: 35Q35.

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